What are all the winning the coalitions in the voting system [25:4,4,6,7,9]?

Question

Write out all the winning coalitions in the voting system [25:4,4,6,7,9]

Let's label each voter, A,B,C,D, and E so the weight of voter A=4, B=4, C=6 and so on.

We have to think about the groups we can make with all the combinations of each voter. There can be groups of 1 voter, groups of 2 voters, 3 voters, group of 4 and 1 group of 5 voters because we do not want to repeat the members in each group.

Answer 1

We need to make all the coalitions or combinations of all the groups and see if the quota is reached by the various combinations of voters. We can make this list by making of groups of 2, 3, 4, and 5.

Group of 5 ABCDE=30 Winning coalition

Groups of 4 ABCD= ABCE=21 ABDE=20 BCDE=26 ACDE= 26

Groups of 3 None of these combined equal or exceed the quota

Groups of 2 None of these combined equal or exceed the quota

Groups of 1= A=4, B=4, C=6, D=7, E=9 none of these reach a quota of 25 by themselves.

The only combinations of voters that equals or exceeds the quota of 25 are ABCDE, BCDE, and ACDE. Therefore, these are the winning coalitions.